Random Matrices in Three Short Stories – II

This is the second part of the series – the first one can be found here.


If you ever happen to be in a conversation with someone with a devout love for number theory, it won’t be long before the conversation will evolve into one about prime numbers – about how fascinating these elementary, yet mystical creatures are and about how despite their apparent randomness, there is an intriguing rhythm, a captivating music in their distribution.

Prime numbers are the prime ingredients of natural numbers (and of interesting conversations, too). Take any natural number (greater than one) – you can always write it as a product of certain primes. And these primes themselves cannot be expressed as the product of smaller numbers. Simple as they sound from this definition, prime numbers have a very surprising and inexplicable manner of showing up on the number line. Much of folklore and mathematical literature alike have their origin in this mystery surrounding primes.

But hold on, why are we talking about primes in a story about random matrices?

We wouldn’t have been, had it not been for a chance teatime conversation between Freeman Dyson and mathematician Hugh Montgomery (interesting conversations, remember) [1].

In the spring of 1972, Montgomery was visiting the Institute for Advanced Study at Princeton, New Jersey to discuss his recent work on the zeros of the Riemann zeta function with fellow mathematician Atle Selberg. Selberg happened to be a leading figure of the time on the Riemann zeta function and the much fabled Riemann Hypothesis.

First formulated by Georg Friedrich Bernhard Riemann in his 1859 paper, the Riemann hypothesis is a hugely famous and celebrated conjecture yet to be (dis)proved. (And guess what, this was the only number theory paper the mathematician extraordinaire Riemann wrote in his entire lifetime!)

Bernhard Riemann made an important observation that the distribution of primes was intricately related to the properties of a function that now bears his name (it was first introduced by the Swiss mathematician Leonhard Euler, though). Now, as it turns out (and as you’ll see if you happen to delve more into abstract mathematics), mathematical functions and objects have personalities of their own. The zeta function, as Riemann observed, appeared to have an interesting one.

\zeta{(s)} = \frac{1}{1^{s}} + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + ...
Behold the mighty Riemann zeta function!
The Riemann Hypothesis states that all the interesting values of s
for which ζ(s) = 0 lie on a straight line in the complex plane.

Numbers as you probably know, can have two parts – a real part and an imaginary one (and you’ll probably realize later that the imaginary part is not so imaginary after all).

a + ib
A complex number has two parts – real and imaginary (here, a and b respectively).
The tiny i hanging alongside the imaginary part, b is what makes it the imaginary part.
(is the imaginary unit, the square root of 1.)

The Riemann zeta function has some non-interesting zeros (values of s for which ζ(s) = 0) at s = -2, -4, -6 and so on. What Riemann conjectured was that all the other zeros of the function will always have their real parts equal to 1/2 – and hence, when you plot them on the complex plane, they’ll all lie on a vertical line [2].

Now, over a century and a half later, all we know is that this hypothesis appears to be true. We know it to be true for the first 1013 (!) zeros we’ve found till now, but have no idea whether it holds true in general or not. (For those of you who refuse to take abstract concepts arising in pure mathematics seriously, the zeta function will keep appearing in your life even if you restrict yourself to more concrete (?) areas of applied mathematics and/or physics.)

As has been the norm with challenging problems since the beginning of the previous century, the Riemann Hypothesis along with five other open problems are listed as the Millennium prize problems by the Clay Mathematics Institute – each with a bounty of a million dollars.

But this story is not about the million dollars – mathematicians don’t care much about it anyway (or so I am guessing). In the early 70s, number theorist Hugh Montgomery was working on the statistical distribution of the interesting zeros of the Riemann zeta function on the critical line – the vertical line where all of them are conjectured to lie on.

When, during their teatime conversation at the Institute for Advanced Study, Montgomery mentioned his recent results to Freeman Dyson, both Dyson and Montgomery were up for a surprise. Dyson realized that the statistical distribution of the Riemann zeros that Montgomery had worked out had a lot in similar with the statistical properties of a certain class of Random Matrices Dyson had earlier looked at while working on the physics of heavy atoms. And more importantly, the theory of random matrices had, by then, pretty well established results that could be applied to Montgomery’s problem [1].

Dyson then wrote a letter to Selberg referring Madan Lal Mehta’s book on random matrices to be looked up for the results that were needed by Montgomery. (You must read this article published in the IAS Spring 2013 newsletter; the article also has a scanned image of Dyson’s handwritten note to Selberg!)

This striking similarity in the statistics of the Riemann zeros and the spectra of heavy atoms points towards an universality in the underlying structures. Stronger results coming from probability theory and mathematical statistics appear to give a clearer picture; though much of it still appears to be an outright miracle. More on this notion of universality in the last part of the series when I’ll narrate the story of one of the greatest quests in present day science – the quest for a theory of quantum gravity.


[1] Kelly Devine Thomas, From Prime Numbers to Nuclear Physics and Beyond, The Institute Letter Spring 2013.
[2] Peter Sarnak, Problems of the Millennium: The Riemann Hypothesis, 2005.

Thanks are due to Divya Singh and Ramesh Chandra for providing essential feedback for this series of posts on Random Matrices. The final post will soon be up.

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